Impact of Inclined Magnetic Field of Casson Fluid on Double Diffusive Natural Convection in a Curvilinear Enclosure

In this paper, the Casson fluid model and the inclined magnetic field are used to analyse mathematically the energy transport of double diffusive natural convection (DDNC) in a curvilinear enclosure. The Galerkin finite element method (GFEM) is used to discretize the equations. When simulating the different ranges of Rayleigh numbers (105 ≤ Ra ≤ 108) and the heated part, we adopt isotherms and streamlines to visualize the thermal and flow fields. In addition, we study the impact of the Casson parameter on the Nusselt number along the cavity wall. The results show that the lower values of Casson fluid parameter have a significant effect on the Nusselt number in the middle of the cavity. Finally, the results are verified with the existing studies, and various parameters such as streamline, isotherm, isoconcentration contour and kinetic energy are presented in the form of graphs and tabular formats.


Introduction
The study of non-Newtonian fluids is crucial in various fields, including biomedical, environmental, and material sciences.These fluids exhibit non-Newtonian behaviour and are used in pharmaceuticals, multiphase mixtures, chemical engineering, lubricants, paints, and food products.Even biological and environmental fluids fall under this category.Compared to Newtonian fluids, mathematical descriptions of non-Newtonian fluids are more complex and involve higher-order equations.Significant advancements have been made by using diverse models to describe the behaviour of these fluids [1].Despite the Casson fluid is recognized as one of the most prominent non-Newtonian fluids, it has not received sufficient attention in prior research studies, which indicates that there is a potential research gap in the field.Several researchers [2][3][4] have conducted extensive studies on Casson fluid, delving into its various properties and behaviour under different conditions for different enclosures.Casson fluid exhibits non-linear, shear-thinning behaviour and yield stress properties.Due to its yield stress property, many researchers studied the behaviour of Casson fluid for the steady and oscillatory blood flow between two rotating cylinders [5] and over an unsteady stretching surface [6].
Magnetohydrodynamics (MHD) studies the interaction between electrically conductive fluids and magnetic fields.MHD is important in designing high-performance pumps, power generators, and cooling systems for industrial applications.Its study is significant in various fields of science and engineering for a better understanding of natural and artificial systems.Krishna [7] conducted a study that involved analysing the parameters and slip conditions of MHD flow of Casson liquid over a porous layer that is deformable, using numerical methods.Rasool et al. [8] examined the Casson fluid flow over a stretched surface in a porous medium with magnetic effects.Kandasamy et al. [9] explored mass and heat transfer in MHD flow over a stretching surface, considering the heat source and thermal stratification effects.
From the above overview, it can be observed that previous studies mostly focused on fluid flow in enclosures, with limited research on thermosolutal diffusive mechanisms in liquids.However, researchers are increasingly interested in diffusive convective transport for various applications.This study aims to explore heat and mass transport in a non-Newtonian Casson liquid in a curvilinear cavity with isothermal and isoconcentration side walls.The paper first reviews existing studies on the fluid model, then presents the problem formulation and computational approach, and finally discusses the results and conclusions.

Mathematical model
The present study focuses on a two-dimensional curvilinear cavity filled with Casson fluid (non-Newtonian), as depicted in Figure 1.The left and right walls are maintained at high and low temperatures and concentrations, respectively, with no-slip boundary conditions assumed for all walls.The fluid properties are considered to be constant except for density, which is analyzed using the Boussinesq model for concentration and temperature.The natural convection flow is governed by the conservation of momentum, concentration and energy in two dimensions through the following equations.:

Validation of results and Grid independence study
To ensure the accuracy of our results, we compared them with those published by Shafqat et al.in [10].
We calculated the variations in the Casson fluid parameter β and the corresponding the averaged Nusselt number while keeping  .,  ,    ,  . , as shown in Table 1a.
where, are fluid density, kinematic viscosity and force terms (magnetic field and Boussinesq approximation).The force includes thermal diffusivity ϱ and solutal expansions .
D is magnetic diffusivity and ς is diffusion coefficient and all the nomenclature presented in [10] [10] Introducing the following dimensionless parameters: The conditions at Low temperature wall are At θ C 0 and U V 0 The conditions at high temperature wall are At θ C 1 and U V 0 The conditions at remaining (adiabatic) walls are At θ 1, C 0 and U V 0 The averaged Nusselt number, averaged Sherwood number and total Kinetic energy can be computed, The Casson parameter, β, is crucial for describing the additional viscosity of the non-Newtonian fluid.Figure 2 shows that when increasing the Casson parameter decreases the liquid viscosity, with low values indicating a significant impact on viscosity.The isotherm patterns reveal that larger Casson fluid parameters β result in more significant interactions between thermal boundary layers, particularly for partially heated side walls.Additionally, increasing the Casson fluid parameter decreases the thickness of the velocity and thermal boundary layers.Figure 3 shows that natural convection problems exhibit significant changes in streamlines, isotherms, and isoconcentration with the Rayleigh number varies.At low Rayleigh numbers, the flow is mainly governed by conductive heat transfer, resulting in small, symmetric streamlines.However, with Rayleigh number increases, natural convection becomes more dominant, leading to the formation of bullous patterns in the cavity.Moreover, the thickness of thermal boundary layer near the side walls decreases with higher Rayleigh numbers due to buoyancy forces' impact.The isotherm behavior becomes non-uniform, and isoconcentrations lines demonstrate intensified convective flow with the increase of Rayleigh number.It can be seen that with the increase of β, γ, Ra and Ha the total kinetic `energy, averaged Nusselt number and averaged Sherwood number increases.when the Lewis number increases, the averaged Nusselt number, and total energy increase but averaged Sherwood number decreases.
Figure 4 shows the variation of the Casson parameter on heat transfer rate, mass transfer rate and total energy.It can be seen that with the increase of the Casson parameter the averaged Nusselt number, averaged Sherwood number and total energy increase but with the increase of Hartman number the Nu , Sh and K.E decreases.

Figure 1 .
Figure 1.Schematic of Curvilinear cavity under magnetic field.
into equations (1-5), the corresponding dimensionless forms and boundary conditions (non-dimension) are

Figure 3 .
Figure 3.The influence of Rayleigh number on flow patterns, Isotherms and Isoconcentrations

Figure 4 .
Figure 4. Variation of average Nusselt number for various Casson parameter ? .
In this study investigates DDNC energy transport in a curvilinear enclosure with Casson fluid and an inclined magnetic field.The Casson parameter significantly impacts the Nusselt number, while increased Casson fluid parameter decreases the thicknesses of velocity and thermal boundary layers.The effect of various parameters, such as Ra, Ha and Le on flow and thermal fields and Nusselt and Sherwood numbers were studied.The flow behavior changes significantly with Rayleigh number, and higher Lewis numbers lead to a rise in fluid concentration near the heated surface.The findings have practical applications in optimizing heat transfer systems in industrial settings.

Table 1b .
Comparison with previous studyIn order to reduce the impact of truncation errors, a grid independence test is conducted and the corresponding numerical values are presented in Table1b.The code's ability to operate independently of the grid size is tested by calculating with various mesh sizes.

Table 2 .
Variation of averaged Nusselt number, averaged Sherwood number and total Kinetic energy for different parameters.

Table 2
represents the mass transfer rate, heat transfer rate and total energy for different parameter like Casson parameter 0.1 β 5 , Inclined angel 0 γ 60 , Rayleigh number 1e Ra 1e , Lewis number 1 Le 10 , and Hartman number 0 Ha 100 .